A sphere inscribed in a cylinder A sphere is said to be inscribed in a cylinder if it touches its base and lateral surface (touches each generatrix). At
Sphere and ball A sphere is the set of all points in space that are at a given distance from a given point. Point O is called the center of the sphere. Any segment connecting the center of the sphere with any point on the sphere is called the radius of the sphere (R). The straight line AB is called the axis, and the points A and B of its intersection with the sphere are the poles of the sphere. A chord of a sphere is a segment connecting two points of a sphere (KN). The diameter of a sphere is a chord passing through its center (AB) R N K
Ball A ball with a center at point O and radius R is the set of all points in space located from point O at a distance not exceeding R. A ball is a body bounded by a sphere. A ball is formed by rotating a semicircle about its fixed diameter (AB). This diameter is called the axis of the ball, and both ends of the specified diameter are the poles of the ball. The surface of a ball is called a sphere. R A B
The part of a ball (sphere) cut off from it by some plane (ABC) is called a spherical segment. Circle ABC is called the base of the spherical segment. The perpendicular segment MN drawn from the center N of circle ABC to the intersection with the spherical surface is called the height of the spherical segment. Point M is called the vertex of the spherical segment. Ball segment Formula: V=1/3P 2 H(3R-H)
Spherical layer The part of the sphere enclosed between two parallel planes ABC and DEF intersecting the spherical surface is called spherical layer. The curved surface of the spherical layer is called spherical belt. Circles ABC and DEF are the bases of the spherical belt. The distance NK between the bases of the spherical belt is its height.
A sphere inscribed in a cone A sphere is said to be inscribed in a cone if it touches all the constituents of the cone and its base. You can fit a sphere into any cone. The center of the sphere lies on the axis of the cone and is the center of a circle inscribed in the axial section of the cone. Formulas for the radius of a ball inscribed in a cone: R - radius of the inscribed ball, r - radius of the base of the cone, l - length of the cone's generatrix, H - height of the cone, A - angle of inclination of the cone's generatrix to its base. l H l r Formulas: R=rtgA/2 R=Hr/(l+r) L r R R O1 A A/2
Problem 1 Problem 1. A ball of radius r is inscribed in a cone. Find the volume of the cone if its height is h. Solution: The axial section of this combination of ball and cone is an isosceles triangle PAB, circumscribed around a circle with center O and radius R, PC = h – height of the cone, OD PB. Volume of the cone Since therefore or whence Therefore, Answer:
Problem 2 A cone of height N is inscribed into a ball of radius R. Find the angle between the generatrix of the cone and the plane of the base. Consider the diametrical section of the ball, as shown in Figure b). As you know, the angle between a straight line and a plane is the angle between this straight line and its projection onto this plane. In our case, AB is a direct line, and AP is a projection. OR = BP-OV = H-R (where H is the height of the cone, R is the radius of the sphere) From the right triangle OAR, we determine the leg AR using the Pythagorean theorem: R H Answer: O
Konas Konas is a body obtained by combining all rays emanating from one point (the vertex of the konas) and passing through a flat surface. Sometimes a konas is a part of such a body obtained by combining all the segments connecting the vertex and points of a flat surface (the latter in this case is called the base of the konas, and the konas is called resting on this base). If the base of the konas is a polygon, the konas becomes a pyramid. A geometric body created by rotating a right triangle around one of its legs
Elements and parts of a konas The vertex is a point at a fixed acute angle of a rotating right triangle forming a konas. The base is a circle bounding the cone, described by the movable leg of the forming triangle. The height of a segment perpendicular to the base, passing through the vertex, the fixed leg of the forming triangle, as well as the length of this segment. Forming a segment connecting the vertex and a point on the circle bounding the base, the hypotenuse of the circumscribing triangle. The lateral surface is a conical surface bounding the cone, formed by the hypotenuse of the generating triangle. o p LATERAL SURFACE FORMING THE BASE OF THE CONE RADIUS APEX AXIS
Truncated cone A truncated cone is a body of rotation formed by the rotation of a rectangular trapezoid near the side perpendicular to the bases. Circles O and O1 are its bases, its constituents AA1 are equal to each other, straight line OO1 is the axis, segment OO1 is the height. Its axial section is an isosceles trapezoid.
Related definitions A segment dropped perpendicularly from the top to the plane of the base (as well as the length of such a segment) is called the height of the cone. The straight line connecting the top and the center of the base is called the axis of the cone. Circular konas a konas whose base is a circle. A cone resting on an ellipse, parabola, or hyperbola is called an elliptic, parabolic, and hyperbolic cone, respectively (the latter two have infinite volume). The part of the cone lying between the base and a plane parallel to the base and located between the top and the base is called a truncated cone.
A cone inscribed in a circle A ball is called circumscribed about a polyhedron, and a polyhedron inscribed in a ball if the surface of the ball passes through all the vertices of the polyhedron. A ball is called circumscribed about a truncated cone (cone) if the circles of the bases (base circle and vertex) belong to the surface of the ball. The center of a ball circumscribed about a polyhedron lies at the intersection point of planes perpendicular to all edges of the polyhedron and passing through their midpoints. It can be located inside, on the surface or outside the polyhedron. A cone is inscribed in a sphere (a sphere is described around a cone) if its vertex belongs to the sphere and its base is a section of a sphere (AOC) bounded by a given sphere. A sphere can always be described around a cone. Its center lies on the axis of the cone and coincides with the center of the circle described around the triangle, which is the axial section of the cone. A B AC O Formulas: R 2 =(H-R) 2 +r 2 R-radius of the ball r-radius of the base of the cone H-height of the cone
\[(\Large(\text(Cylinder)))\]
Consider a circle \(C\) with center \(O\) of radius \(R\) on the plane \(\alpha\) . Through each point of the circle \(C\) we draw a straight line perpendicular to the plane \(\alpha\) . The surface formed by these straight lines is called cylindrical surface.
The straight lines themselves are called forming of this surface.
Let us now draw a plane \(\beta\parallel \alpha\) through some point of some generator. The set of points along which the generators intersect the plane \(\beta\) forms a circle \(C"\) equal to the circle \(C\) .
A part of space bounded by two circles \(K\) and \(K"\) with boundaries \(C\) and \(C"\), respectively, as well as a part of a cylindrical surface enclosed between the planes \(\alpha\) and \(\beta\) , called cylinder.
Circles \(K\) and \(K"\) are called the bases of the cylinder; the segments of the generatrices enclosed between the planes are the generators of the cylinder; the part of the cylindrical surface formed by them is the lateral surface of the cylinder. The segment connecting the centers of the bases of the cylinder is equal to the generatrix of the cylinder and equal to the height of the cylinder (\(l=h\) ).
Theorem
The lateral surface area of the cylinder is equal to \
where \(R\) is the radius of the base of the cylinder, \(h\) is the height (generative).
Theorem
The total surface area of a cylinder is equal to the sum of the area of the lateral surface and the areas of both bases \
Theorem
The volume of the cylinder is calculated by the formula \
\[(\Large(\text(Cone)))\]
Consider the plane \(\alpha\) and on it a circle \(C\) with center \(O\) and radius \(R\) . Through the point \(O\) we draw a straight line perpendicular to the plane \(\alpha\) . Let us mark some point \(P\) on this line. The surface formed by all lines passing through the point \(P\) and each point of the circle \(C\) is called conical surface, and these straight lines are the generators of the conical surface. The part of space bounded by a circle with boundary \(C\) and segments of generators enclosed between the point \(P\) and a point on the circle is called cone. The segments \(PA\) , where \(A\in \text(env. ) C\) , are called forming a cone; point \(P\) – vertex of the cone; circle with boundary \(C\) – base of the cone; segment \(PO\) – height of the cone.
Comment
Note that the height and generatrix of a cone are not equal to each other, as was the case with a cylinder.
Theorem
The lateral surface area of the cone is equal to \
where \(R\) is the radius of the base of the cone, \(l\) is the generator.
Theorem
The total surface area of the cone is equal to the sum of the lateral surface area and the base area \
Theorem
The volume of the cone is calculated by the formula \
Comment
Note that the cylinder, in a sense, is a prism, only at the base there is not a polygon (like a prism), but a circle.
The formula for the volume of a cylinder is the same as the formula for the volume of a prism: the product of the area of the base and the height.
Likewise, a cone is in a sense a pyramid. Therefore, the formula for the volume of a cone is the same as that of a pyramid: a third of the area of the base times the height.
\[(\Large(\text(Sphere and ball)))\]
Let's consider a set of points in space equidistant from some point \(O\) at a distance \(R\) . This set is called sphere with center at point \(O\) of radius \(R\) .
A segment connecting two points of a sphere and passing through its center is called the diameter of the sphere.
The sphere together with its interior is called ball.
Theorem
The area of the sphere is calculated by the formula \
Theorem
The volume of the ball is calculated by the formula \
Definition
A spherical segment is a part of a ball cut off from it by a certain plane.
Let the plane intersect the ball in a circle \(K\) with center at the point \(Q\) . Let's connect the points \(O\) (the center of the ball) and \(Q\) and extend this segment until it intersects with the sphere - we get the radius \(OP\) . Then the segment \(QP\) is called the height of the segment.
Theorem
Let \(R\) be the radius of the ball, \(h\) be the height of the segment, then the volume of the spherical segment is equal to \
Definition
A spherical layer is a part of a ball enclosed between two parallel planes intersecting this ball. The circles along which the planes intersect the ball are called the bases of the spherical layer, the segment connecting the centers of the bases is called the height of the spherical layer.
The two remaining parts of the ball are in this case spherical segments.
The volume of the spherical layer is equal to the difference between the volume of the sphere and the volumes of spherical segments with heights \(AP\) and \(BT\).
Solving problems on a cone inscribed in a ball (cone inscribed in a sphere) comes down to considering one or more triangles.
A cone is inscribed in a ball if its vertex and base circumference lie on the surface of the ball, that is, on a sphere. The center of the ball lies on the axis of the cone.
When solving problems involving a cone inscribed in a ball, it is convenient to consider the section of a combination of bodies by a plane passing through the axis of the cone and the center of the ball. The section is a large circle of a ball (that is, a circle whose radius is equal to the radius of the ball) with an isosceles triangle inscribed in it - the axial section of the cone. The sides of this triangle are the forming parts of the cone, the base is the diameter of the cone.
If the angle between the generators is acute, the center of the circumscribed circle lies inside the triangle (correspondingly, the center of the ball circumscribed about the cone is inside the cone).
If the angle between the generators is right, the center of the circle lies in the middle of the base of the triangle (the center of the ball coincides with the center of the base of the cone).
If the angle between the generators is obtuse, the center of the circle lies outside the triangle (the center of the circumscribed ball is outside the cone).
If the problem statement does not say where exactly the center of the circumscribed ball lies, it is advisable to consider how different options for its location may affect the solution.
Consider a cone and a ball circumscribed about it by a plane passing through the axis of the cone and the center of the ball. Here SO=H is the height of the cone, SB=l is the generatrix of the cone, SO1=O1B=R is the radius of the ball, OB=r is the radius of the base of the cone, ∠OSB=α is the angle between the height and the generatrix of the cone.
Triangle SO1B is isosceles with base SB (since SO1=O1B=R). This means that its base angles are equal: ∠OSB=∠O1BS=α, and O1F is the median, height and bisector. Hence SF=l/2.
When solving problems involving a cone inscribed in a ball, you can consider right triangles SFO1 and SOB. They are similar (according to the acute angle S). From the similarity of triangles
In a right triangle SOB ∠OBS=90º - ∠OSB=90º-α. According to the Pythagorean theorem
In the right triangle O1OB ∠OBO1=90º - ∠O1BS=90º - α - α=90º - 2α.
A sphere inscribed in a cone A sphere is called inscribed in a cone if it touches its base and lateral surface (touches each generatrix). In this case, the cone is said to be circumscribed about the sphere. A sphere can be inscribed into any cone (straight, circular). Its center is at the height of the cone, and its radius is equal to the radius of the circle inscribed in the triangle, which is the axial section of the cone. Recall that the radius r of a circle inscribed in a triangle is found by the formula r S p, where S is the area, p is the semi-perimeter of the triangle.
Exercise 3 The radius of the base of the cone is 1. The generatrix is inclined to the plane of the base at an angle of 45°. Find the radius of the inscribed sphere. Solution. The height SH of cone 2 is equal to 1. Generator. 1 The semi-perimeter p is equal to 2. By the formula r = S/p, we have 2 1. 2 1. r 1 1 2 r Answer:
Exercise 4 The height of the cone is 8, forming 10. Find the radius of the inscribed sphere. Solution. The radius of the base of the cone is 6. The area of the triangle SFG is 48, the semi-perimeter is 16. Using the formula r = S/p, we have r = 3. Answer: r = 3.
A sphere circumscribed about a cone A sphere is said to be circumscribed about a cone if the vertex and the circumference of the base of the cone lie on the sphere. In this case, the cone is said to be inscribed in a sphere. Around any cone (straight, circular) you can describe a sphere. Its center is at the height of the cone, and its radius is equal to the radius of the circle described around the triangle, which is the axial section of the cone. Recall that the radius R of a circle circumscribed about a triangle, abc, is found by the formula S 4, where S is the area, a, b, c are the sides of the triangle. R
Exercise 1 A sphere is described around a cone whose base radius is 1 and generatrix is 2. Find its radius. Solution. Triangle SAB is equilateral with side 2. Height SH is equal Area S is equal Using the formula R = abc/4S 3. we obtain 3. R 2 3 3 .
Exercise 2 A sphere of radius 5 is described around a cone whose base radius is 4. Find the height h of the cone. Solution. We have, OB = 5, HB = 4. Therefore, OH = 3. Considering that SO = OB = 5, we get h = 8. Answer: h = 8.
Polyhedra inscribed in a sphere Theorem. A sphere can be described near a prism if and only if a circle can be described near the base of this prism. Its center will be the middle of the segment connecting the centers of the circles described around the bases of the prism. The radius of the sphere R is calculated by the formula point O, which is where h is the height of the prism, r is the radius of the circle circumscribed around the base of the prism. R r 2 , 2 h 2
Exercise 1 Find the radius of a sphere circumscribed about a unit cube. Answer: R 3 2 .
Exercise 2 Find the edge of a cube inscribed in the unit sphere. Answer: a 2 3 3 .
“Inscribed angle” - Given: __A. Repetition of material. Find the mistake in the wording: Knowing how it is expressed. The size of the central angle. The magnitude of the inscribed angle. Problem #1: Compare the size of the external angle and the angle at the base. How are angles AOB and ACB similar and different? According to figure b). find the size of the external angle. Construction of perpendicular lines.
“Measuring angles” - Acute, straight, obtuse, straight angles. Measuring angles. A protractor is used to construct angles. You can attach the protractor differently. Right angle. Obtuse angle. A protractor is used to measure angles. Sharp corner. Unfolded corner. What angle does the hour and minute hands of a clock make?
“Inscribed Angle Theorem” - What is the name of an angle with its vertex at the center of the circle. The concept of an inscribed angle. Find the angle between the chords. Answer. Solution. Inscribed angle theorem. Triangle. Consolidation of the studied material. Sharp corner. Check yourself. Find the angle between them. Correct answer. Updating students' knowledge. Radius of a circle.
“Angle and its measurement” - The hour and minute hands of a clock form an obtuse angle at 5 o’clock. Construction of angles. On checkered paper. Unfolded corner. Obtuse angle. Sharp corner. A protractor is used to measure angles. A right angle is half a turned angle. Measuring angles. Using a protractor. Angles are measured in degrees.
“An angle inscribed in a circle” - Corollaries. Indicate the inscribed angles shown in the figure. Inscribed angle. Which angle is called central? Lesson objectives. An angle whose vertex lies on a circle. Cases of beam location. Find it. An inscribed angle is measured by the half of the arc on which it subtends. Which of the angles shown in the figure are inscribed?
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